# First-order linear differential equation

## Contents

## Definition

### Format of the differential equation

A **first-order linear differential equation** is a differential equation of the form:

where are known functions.

### Solution method and formula: indefinite integral version

Let be an antiderivative for , so that . Then, we multiply both sides by . Simplifying, we get:

Integrating, we get:

Rearranging, we get:

where is an antiderivative of .

In particular, we obtain that:

The function is termed the integrating factor for the differential equation because multiplying by this turns the differential equation into an exact differential equation, i.e., a differential equation to which we can apply integration on both sides.

### Solution method and formula: definite integral version

Suppose we are given the initial value condition that at .

Let be an antiderivative for , so that . Then, we multiply both sides by . Simplifying, we get:

Integrating from to (arbitrary) , we get:

Thus, the general expression is:

## Examples

### Simple example

Consider the differential equation:

Here, . Take and get:

This gives:

### Example that is better solved by subtitution

Consider:

Divide both sides by to get:

This is linear, with , . Take and (see note):

This gives:

The linear method is unnecessary -- we divided and multiplied by . A better solution would be to substitute and get a separable differential equation.

### Example where a particular solution is obtained by inspection

Consider:

The linear method gives:

The integration is not easy. So, instead of trying to do the integration directly, we note that the answer is:

It thus suffices to find a particular solution. Inspection and guesswork gives a solution . The general solution is thus: